Super Particular Stepsize

In: /tuning/topicId_31138.html#31404

Paul wrote:

<< Again, from the ancient Greeks to today, superparticularity was
considered a desirable feature to have *in the ratios describing the
step sizes in the scale*, NOT *in the ratios describing the pitches
in the scale*. >>

Paul,

Thank you for the information! It seems that my mind now asks similar questions as before (where the pitch of scale intervals relative to 1/1 was the item of interest), but now (considering your and Kraig's comments) directed towards the ratiometric "step sizes" between the individual pitches of a scale.

Indeed, once one begins to "rotate" the scale (thus referencing from other individual pitches within a scale), such stepsizes *become* the "interval of interest" (relative to each pitches adjacent neighbors) ...

And stepsizes of stepsizes, etc. Is there any "functional" meaning to the "stepsizes of the stepsizes of the stepsizes" (and onward), until one becomes "comma-tose"? :)

What interests me, in all this ratio stuff, are the spectral implications of the choices of utilizing this ratio or that [relevant to whatever ratiometric level is associated with the calculation and analysis of the resultant (harmonic) spectrums resulting from the potential existence of such overtones], that is - energy appearing at integer multiples of the fundamental frequency of the sin/cos functions (when the frequency of such sin/cos fundamentals is described by rational numbers taken to integer-valued powers in the exponent).

While I can no more mathematically model the composite spectra of my guitar's ringing, sustaining strings sweet tones in time/frequency than any other mortal, recognizing that the entirety of the synthesis, acoustic, and perceptual processes at play evade such "in vitro" atomizations, I would hope, anyway, that this "superparticularity" bit, which many seem to prefer, have at least some sort of grounding possible in theoretical numerical modelling which *might* further enlighten us as to its origins (and, possibly, its applications). This is, admittedly, a fully "left-hemisphere" endeavor for me. With or without the existence of such structures, improvisor that I am, such platitudes will not demarcate what pitches my ear chooses when I play my instrument. It may not yield interesting musical composition on my part. It may just be a pretty diagram, an object of "in vitro" mathematical "worship"... Yet mathematical curiosity drives me - to find out what, if anything, is there.

"Because it's (possibly) there"... If not, I play on, no worse for the wear...

What can we definitively state about the resulting harmonic spectrums resulting from such a (perhaps feeble, perhaps instructive) model of musical spectrums. I find William Sethares' work fascinating, and have much to read and think about in regards to his ideas. Meanwhile, I have been exploring some numerical relationships surrounding the above stated specific interests, and wondered if you (or others) have, in the many hours some have spent pondering matters of (JI) scale intervals (and step sizes, commas, etc.), endeavored to find (comprehensive, rather than anecdotal or isolated) pattern and periodicity - specifically, in the actual *harmonic* spectrums (potentially) resulting from the various combinations of the pitches of a given musical scale(s)?

Maybe I'm just not reading the right folks, but I (in my short career thinking about such mathematical analysis of musical structures, as opposed to the analysis of already existing musical waveforms and signals) do not see much out there which offers much more than a fleeting and "partial" (bad pun) glimpse of the coincident harmonic structures which evolve out of the utilization of rational scale intervals, in general. So I endeavor to (re -) chart the pathways, and find the "golden fleece" of harmonicity (if there is any) :)

There seems to be as much (if not more) stuff on "new-age" and "astrology" websites utilizing the "phraseology" of "harmonics" and "overtones" (most often absurdly, and ignorantly "whoring" these words to any manner of flaky ends) than stuff to be found relating to musical applications which is not largely "Byzantine" in its convoluted, spotty, anecdotal, and narrow focus.
It (almost) always strikes me as "tunneled" and "isolated" information, leaving me feeling like I never know whether begin to try to piece together a number of such "partial pieces" of the pie, or just figure it out for myself as I choose.

Why the (apparent) scarcity of (unified and comprehensive) theories relating to such multi-complex-tone resultant harmonic spectra? Is there no pattern?

David Canright's stuff has been some of the most promising and fruitful found (in regards to clarity and the citation of useful information). Perhaps you or others could make me aware of the existence of the work of others who have attempted to follow such complicated harmonic pathways before, and lived to tell the tale of pattern, structure, symmetry, beauty ... or random chaos devoid of coherent or repeatable structures ...

This is an octave specific world (in addition to being, in other levels of interpretation, octave- invariant). It is a world where the harmonic spectrum of an interval inversion around a given pitch is non-equivalent musically (harmonically, that is). It is not a world which I have seen anyone visiting (but, then again, I have not seen much of the world). Is anybody out there?

Sincerely, J Gill

--- In tuning@y..., J Gill <JGill99@i...> wrote:

> And stepsizes of stepsizes, etc.

These will be the unison vectors of the scale (assuming it's CS to
start with).

> Is there any "functional" meaning to the
> "stepsizes of the stepsizes of the stepsizes"

These will simply be further unison vectors.

> (and onward),

And so on. Stepsizes of stepsizes is about as far as you need to go.
Then you can either add _or_ subtract these to get further unison
vectors.

> until one
> becomes "comma-tose"? :)

LOL!
>
> What interests me, in all this ratio stuff, are the spectral
implications
> of the choices of utilizing this ratio or that [relevant to
whatever
> ratiometric level

What's a ratiometric level?

> is associated with the calculation and analysis of the
> resultant (harmonic) spectrums resulting from the potential
existence of
> such overtones],

Hmm . . . this sort of thing has been done over and over again since
Helmholtz if not earlier, but what is relevant there is of course the
ratios of the intervals _between_ the notes, not the "pitch-ratios"
as such.

> While I can no more mathematically model the composite spectra of
my
> guitar's ringing, sustaining strings sweet tones in time/frequency
than any
> other mortal,

If you take sympathetic vibration into account, you might have a much
better chance. Most synthesizers and samplers fail to mimic guitar
chords well because they simply _add_ the individual samples,
neglecting to model the effects of various partials sympathetically
resonating with one another, effects which can be overwhelming on a
real guitar. This is why _Utonal_ chords sound much better on an
acoustic guitar that on any other instrument I've tried.

> I would
> hope, anyway, that this "superparticularity" bit, which many seem
to
> prefer, have at least some sort of grounding possible in
theoretical
> numerical modelling which *might* further enlighten us as to its
origins
> (and, possibly, its applications).

I think, or at least hope, Kraig and I have given you a sense of this
grounding.

> What can we definitively state about the resulting harmonic
spectrums
> resulting from such a (perhaps feeble, perhaps instructive) model
of
> musical spectrums. I find William Sethares' work fascinating, and
have much
> to read and think about in regards to his ideas. Meanwhile, I have
been
> exploring some numerical relationships surrounding the above stated
> specific interests, and wondered if you (or others) have, in the
many hours
> some have spent pondering matters of (JI) scale intervals (and step
sizes,
> commas, etc.), endeavored to find (comprehensive, rather than
anecdotal or
> isolated) pattern and periodicity - specifically, in the actual
*harmonic*
> spectrums (potentially) resulting from the various combinations of
the
> pitches of a given musical scale(s)?

Sure -- this is a cornerstone of all this research. The simpler the
ratio describing the interval _between_ two pitches, the more
spectral commonality the two pitches will have, and the simpler the
periodic pattern they will produce. Therefore, virtually all my work
in this area has concerned itself with finding scales that have a
very high proportion of simple ratios (or good approximations
thereof) _between_ pitches.

> It (almost) always strikes me as "tunneled" and "isolated"
information,
> leaving me feeling like I never know whether begin to try to piece
together
> a number of such "partial pieces" of the pie, or just figure it out
for
> myself as I choose.

Well, I hope I can be of assitance to you, because I, for one, have
endeavored to "figure it out for myself" from the very beginning,
never for once taking anyone's word for anything, and have now gotten
to the point where I can understand the work of the majority of
historical figures in the field in my own terms.

> Why the (apparent) scarcity of (unified and comprehensive) theories
> relating to such multi-complex-tone resultant harmonic spectra? Is
there no
> pattern?

Perhaps the pattern is far simpler than you have imagined,
particularly when you take virtual pitch into account.

> David Canright's stuff has been some of the most promising and
fruitful
> found (in regards to clarity and the citation of useful
information).
> Perhaps you or others could make me aware of the existence of the
work of
> others who have attempted to follow such complicated harmonic
pathways
> before, and lived to tell the tale of pattern, structure, symmetry,
beauty
> ... or random chaos devoid of coherent or repeatable structures ...

Have you gone through Helmholtz's book?

> It is not a world which I have seen
> anyone visiting (but, then again, I have not seen much of the
world). Is
> anybody out there?

You betcha -- even on this list for the past six years, much of this
world has been visited and mapped out.

--- In tuning@y..., J Gill <JGill99@i...> wrote:

/tuning/topicId_31418.html#31418

>
> What interests me, in all this ratio stuff, are the spectral
implications of the choices of utilizing this ratio or that [relevant
to whatever ratiometric level is associated with the calculation and
analysis of the resultant (harmonic) spectrums resulting from the
potential existence of such overtones], that is - energy appearing at
integer multiples of the fundamental frequency of the sin/cos
functions (when the frequency of such sin/cos fundamentals is
described by rational numbers taken to integer-valued powers in the
exponent).
>
> While I can no more mathematically model the composite spectra of
my guitar's ringing, sustaining strings sweet tones in time/frequency
than any other mortal, recognizing that the entirety of the
synthesis, acoustic, and perceptual processes at play evade such "in
vitro" atomizations, I would hope, anyway, that
this "superparticularity" bit, which many seem to prefer, have at
least some sort of grounding possible in theoretical numerical
modelling which *might* further enlighten us as to its origins
> (and, possibly, its applications).

Hello J!

Well, unless I'm missing something, the ratios are right there, so
you can see the simple relationship with the fundamentals and,
correspondingly, the significance. Aren't you, perhaps, making it a
bit more difficult than it is??

In any case, you must be refering to the construction of *scales*
with this and you're right, William Sethares is the master of this
realm. I do suggest you read his _Tuning Timbre Spectrum Scale_ if
you haven't already (it wasn't quite clear in your post)... It's
exactly up your overtone series!

Joseph Pehrson

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> Hello J!
>
> Well, unless I'm missing something, the ratios are right there, so
> you can see the simple relationship with the fundamentals and,
> correspondingly, the significance. Aren't you, perhaps, making it
> a bit more difficult than it is??

JG: Perhaps, Joe... When considering the harmonic terrain resulting
from the combination of each of a scales' intervals with all of the
other intervals which make up the scale [as well as the cases of the
simultaneous combinations of two (or a larger number of) intervals
within that group], my goal is to define potential structures of
harmonic interplay (in a unified and understandable manner) which
*may* (in part) contribute to various choices of JI scale intervals
(however arrived at), thus (hopefully) lending insight into why "this
or that" combination of pitches (the inter-tone "ratiometric
intervals", as referenced to a pitch of 1/1, or as refernced to any
of the other pitches existing as a member of a "playable gamut" of
pitches under consideration) are found (by some) to be desirable
musical choices.

I myself do not find that the quantity which is commonly referred to
as "harmonic distance" [that is: (numerator) * (denominator)] to be
an accurate "metric" of "harmonic coincidence". As a result, when
modelling the frequency locations of "harmonic coincidence" existing
between two (or more) scale intervals sounded simultaneously, I
utilize a different numerical method for determining such results.

> In any case, you must be refering to the construction of *scales*
> with this

JG: I am seeking to define a central "gamut" of possible notes
(through the prime 7-limit) which form a basis set which is made up
of certain JI intervals which possess commonalities in the manner in
which they interact with other members of the set where "harmonic
coincidence" (and its musical implications) is concerned. (As I see
it), in order for a unified harmonic/melodic JI structure to exist,
certain sets of JI intervals and the (melodic) implications of their
inclusion within certain (N-tone, where N is an even valued number)
JI scales should reflect what constitutes (as a result of their
inclusion within that central "gamut") a coherent basis in the
(harmonic) relationships contained within the "universe" of intervals
included in that central "gamut" of (potential) JI scale intervals.

If you (or anyone else out there) know of any "prior-art" which
similarly addresses the above stated goal [of successfully achieving
a "unification" of certain JI scale intervals (or "melodic" choices)
utilizing scale intervals derived from within the "universe" of such a
(harmonically derived) central "gamut" of (potential) JI scale
intervals], I would be *very* interested in hearing about your (or
their) thoughts and ideas regarding such relationships (of which, in
my very limited knowledge of, and exposure to, the work of others, I
am not, as yet, aware of as existing). I really dig this stuff! :)

This endeavor of mine is *not* about tempering intervals in order to
make instrument tunings maximally usable in the case of physical
movement of patterns across a keyboard (or the equivalent tunings on
other instruments), freeing one from the (no less real) complications
of rotating the reference pitch in any (fixed) JI instrument tuning,
and instead, allowing concentration on (possible) harmonic-based
relationships from which the (melodic) choice of specific scale
intervals [at least "partially" (bad pun)] derive. It does not
appear that values of the step-sizes between each of the ascending
scale intervals represents more than a *portion* of the information
of interest necessary in order to pursue the latter endeavor.

It *is* about why certain JI intervals *may* [when a (complex) note
is simultaneously sounded in unison with, and relative to, any
additional given (complex) interval(s) and associated timber(s)
existing within a central "gamut" of (potential) JI scale intervals],
demonstrate various levels of tonal affinity for each other, as well
as demonstrate definable similarities between certain (complex)
intervals in terms of their characteristic harmonic interactions with
other (complex) intervals which exist within the "universe" of
intervals included within a central "gamut" of (potential) JI scale
intervals.

> and you're right, William Sethares is the master of this
> realm. I do suggest you read his _Tuning Timbre Spectrum Scale_ if
> you haven't already (it wasn't quite clear in your post)... It's
> exactly up your overtone series!

JG: I have yet to do more than briefly look over a couple of Mr
Sethares' papers available on the internet. Reading about him does,
indeed, call forth a sense of a close "affinity" existing between our
individual areas of interest, with Mr Sethares far advanced (in
theoretical understanding, resultant thought performed about such
related issues, and so on) relative to my humble, largely self-
taught, and largely self-directed personal studies and speculations.
I, no doubt, would *love* to be his student (given the fascinating
areas of his exploration), and may well treat myself to a copy of
his "Tuning_Timbre_Spectrum_Scale" book, which, indeed looks to
be "right up my alley", alright! Is there anything in his book which
resembles my above (I hope, adequately described) tonal endeavors?

It's very nice chatting with you, my friend! :)

Sincerely, J Gill

--- In tuning@y..., "unidala" <JGill99@i...> wrote:

> JG: Perhaps, Joe... When considering the harmonic terrain resulting
> from the combination of each of a scales' intervals with all of the
> other intervals which make up the scale [as well as the cases of
the
> simultaneous combinations of two (or a larger number of) intervals
> within that group], my goal is to define potential structures of
> harmonic interplay (in a unified and understandable manner) which
> *may* (in part) contribute to various choices of JI scale intervals
> (however arrived at), thus (hopefully) lending insight into
why "this
> or that" combination of pitches (the inter-tone "ratiometric
> intervals", as referenced to a pitch of 1/1, or as refernced to any
> of the other pitches existing as a member of a "playable gamut" of
> pitches under consideration) are found (by some) to be desirable
> musical choices.

Well, that's pretty close to my own philosophy of scale construction,
and how I understand most of the previous attempts.

> I myself do not find that the quantity which is commonly referred
to
> as "harmonic distance" [that is: (numerator) * (denominator)] to be
> an accurate "metric" of "harmonic coincidence". As a result, when
> modelling the frequency locations of "harmonic coincidence"
existing
> between two (or more) scale intervals sounded simultaneously, I
> utilize a different numerical method for determining such results.

Could you elaborate it?
>
> > In any case, you must be refering to the construction of *scales*
> > with this
>
>
> JG: I am seeking to define a central "gamut" of possible notes
> (through the prime 7-limit)

Odd 7-limit, perhaps?

> which form a basis set which is made up
> of certain JI intervals which possess commonalities in the manner
in
> which they interact with other members of the set where "harmonic
> coincidence" (and its musical implications) is concerned.

Sounds like a description of the 7-limit Tonality Diamond to me, if
you're centering around a 1/1. If not, the Hexany or Stellated Hexany
may fit the bill.

> (As I see
> it), in order for a unified harmonic/melodic JI structure to exist,
> certain sets of JI intervals and the (melodic) implications of
their
> inclusion within certain (N-tone, where N is an even valued number)
> JI scales

Why should N be even?

> should reflect what constitutes (as a result of their
> inclusion within that central "gamut") a coherent basis in the
> (harmonic) relationships contained within the "universe" of
intervals
> included in that central "gamut" of (potential) JI scale intervals.

Agreed!

> If you (or anyone else out there) know of any "prior-art" which
> similarly addresses the above stated goal [of successfully
achieving
> a "unification" of certain JI scale intervals (or "melodic" choices)

Well if melody comes into the picture, I'd tend to embed the
structure of interest within a periodicity block -- lots of 31-tone
examples fit the bill nicely here.

> utilizing scale intervals derived from within the "universe" of
such a
> (harmonically derived) central "gamut" of (potential) JI scale
> intervals], I would be *very* interested in hearing about your (or
> their) thoughts and ideas regarding such relationships (of which,
in
> my very limited knowledge of, and exposure to, the work of others,
I
> am not, as yet, aware of as existing).

Have you read Partch's book?

> and instead, allowing concentration on (possible) harmonic-based
> relationships from which the (melodic) choice of specific scale
> intervals [at least "partially" (bad pun)] derive. It does not
> appear that values of the step-sizes between each of the ascending
> scale intervals represents more than a *portion* of the information
> of interest necessary in order to pursue the latter endeavor.

Of course.

> It *is* about why certain JI intervals *may* [when a (complex) note
> is simultaneously sounded in unison with, and relative to, any
> additional given (complex) interval(s) and associated timber(s)
> existing within a central "gamut" of (potential) JI scale
intervals],
> demonstrate various levels of tonal affinity for each other, as
well
> as demonstrate definable similarities between certain (complex)
> intervals in terms of their characteristic harmonic interactions
with
> other (complex) intervals which exist within the "universe" of
> intervals included within a central "gamut" of (potential) JI scale
> intervals.

You're telling me you haven't read Partch's book?

--- In tuning@y..., "unidala" <JGill99@i...> wrote:

> I myself do not find that the quantity which is commonly referred
to
> as "harmonic distance" [that is: (numerator) * (denominator)] to be
> an accurate "metric" of "harmonic coincidence".

I've been calling it "Tenney height", since it is what number
theorists call a height function, but if "harmonic distance" is a
name people understand as meaning this that could change. What *do*
you consider to be an accurate metric?

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> > I myself do not find that the quantity which is commonly referred
> to
> > as "harmonic distance" [that is: (numerator) * (denominator)] to
be
> > an accurate "metric" of "harmonic coincidence".
>
> I've been calling it "Tenney height", since it is what number
> theorists call a height function, but if "harmonic distance" is a
> name people understand as meaning this that could change. What *do*
> you consider to be an accurate metric?

Sethares himself, for the simple ratios and a generic harmonic
timbre, seems to derive a "sensory dissonance" measure very much in
agreement with n*d or n+d.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> > I myself do not find that the quantity which is commonly referred
> to
> > as "harmonic distance" [that is: (numerator) * (denominator)] to
be
> > an accurate "metric" of "harmonic coincidence".
>
> I've been calling it "Tenney height", since it is what number
> theorists call a height function, but if "harmonic distance" is a
> name people understand as meaning this that could change.

Since it's applied to the interval _between_ two pitch-
ratios, "distance" would seem to make more sense -- especially in
conjunction with the spacial representation that is the Tenney
lattice.

> What *do*
> you consider to be an accurate metric?

Helmholtz also noted an n*d ordering from his results. Benedetti in
the 16th century came to the same conclusion. By imposing octave-
equivalence, I find that an odd-limit condition is implied, which
agrees perfectly with Partch's independent finding.

--- In tuning@y..., "unidala" <JGill99@i...> wrote:

/tuning/topicId_31418.html#31484
> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
>
> JG: I am seeking to define a central "gamut" of possible notes
> (through the prime 7-limit) which form a basis set which is made up
> of certain JI intervals which possess commonalities in the manner
in
> which they interact with other members of the set where "harmonic
> coincidence" (and its musical implications) is concerned. (As I see
> it), in order for a unified harmonic/melodic JI structure to exist,
> certain sets of JI intervals and the (melodic) implications of
their
> inclusion within certain (N-tone, where N is an even valued number)
> JI scales should reflect what constitutes (as a result of their
> inclusion within that central "gamut") a coherent basis in the
> (harmonic) relationships contained within the "universe" of
intervals
> included in that central "gamut" of (potential) JI scale intervals.
>
> If you (or anyone else out there) know of any "prior-art" which
> similarly addresses the above stated goal [of successfully
achieving
> a "unification" of certain JI scale intervals (or "melodic"
choices)
> utilizing scale intervals derived from within the "universe" of
such a
> (harmonically derived) central "gamut" of (potential) JI scale
> intervals], I would be *very* interested in hearing about your (or
> their) thoughts and ideas regarding such relationships (of which,
in
> my very limited knowledge of, and exposure to, the work of others,
I
> am not, as yet, aware of as existing). I really dig this stuff! :)
>

Huh? But isn't that what Harry Partch was trying to do with his
tonality diamond?? Isn't that what he was getting at?? Good stuff
there, that Harry Partch... Even Jon Szanto will agree to that.

JP

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
>
> /tuning/topicId_31418.html#31484
> > --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> >
> >
> > JG: I am seeking to define a central "gamut" of possible notes
> > (through the prime 7-limit) which form a basis set which is made
up
> > of certain JI intervals which possess commonalities in the manner
> in
> > which they interact with other members of the set where "harmonic
> > coincidence" (and its musical implications) is concerned. (As I
see
> > it), in order for a unified harmonic/melodic JI structure to
exist,
> > certain sets of JI intervals and the (melodic) implications of
> their
> > inclusion within certain (N-tone, where N is an even valued
number)
> > JI scales should reflect what constitutes (as a result of their
> > inclusion within that central "gamut") a coherent basis in the
> > (harmonic) relationships contained within the "universe" of
> intervals
> > included in that central "gamut" of (potential) JI scale
intervals.
> >
> > If you (or anyone else out there) know of any "prior-art" which
> > similarly addresses the above stated goal [of successfully
> achieving
> > a "unification" of certain JI scale intervals (or "melodic"
> choices)
> > utilizing scale intervals derived from within the "universe" of
> such a
> > (harmonically derived) central "gamut" of (potential) JI scale
> > intervals], I would be *very* interested in hearing about your
(or
> > their) thoughts and ideas regarding such relationships (of which,
> in
> > my very limited knowledge of, and exposure to, the work of
others,
> I
> > am not, as yet, aware of as existing). I really dig this
stuff! :)
> >
>
>
> Huh? But isn't that what Harry Partch was trying to do with his
> tonality diamond?? Isn't that what he was getting at?? Good stuff
> there, that Harry Partch... Even Jon Szanto will agree to that.
>
> JP

JG: Indeed (from the little that I do understand about Partch's
writings, *not* having read his book, and from "peeking" at some
treatments by other persons - mostly folks in this group), it seems
that my goals are certainly nothing new or unique!... :)

My methods (in certain respects) for "justaposing" (bad pun) scale
interval ratios, and "categorizing" such interval ratios appears
(from my limited exposure to Partch's concepts) to be different from
what Partch appears to have done. Had for this not been the case, I
would (if anything) probably be doing *more* reading of Partch and
related/derivative work, and "less* "nerding-out" on my own ideas.

Both can be valuable (though time consuming for an average intellect
such as mine)! I do wish that I had more time to do both!... :)

If, upon (a comprehensive and coherent) presentation of my ideas
(which I would like to be able to do), I eventually were to discover
that there appeared to be *no* other before me who has grappled with
these mathematical relationships, it would (I am sure) be a miracle!

Best Regards, J Gill

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning@y..., "unidala" <JGill99@i...> wrote:
> >
> > > I myself do not find that the quantity which is commonly
referred
> > to
> > > as "harmonic distance" [that is: (numerator) * (denominator)]
to
> be
> > > an accurate "metric" of "harmonic coincidence".
> >
> > I've been calling it "Tenney height", since it is what number
> > theorists call a height function, but if "harmonic distance" is a
> > name people understand as meaning this that could change.

J Gill: Paul, I guess I got that phrase from your ATL message #26247:

PE: "Sure. Two notes having a musical ratio a:b, have Tenney Harmonic
Distance log(a*b) = log(a) + log (b). Logs to the base 2 are used, so
if you need to calculate these, use
log(a*b)/log(2)
or
log(a)/log(2) + log(b)/log(2)"

> Since it's applied to the interval _between_ two pitch-
> ratios, "distance" would seem to make more sense -- especially in
> conjunction with the spacial representation that is the Tenney
> lattice.
>
> > What *do*
> > you consider to be an accurate metric?
>
> Helmholtz also noted an n*d ordering from his results. Benedetti in
> the 16th century came to the same conclusion. By imposing octave-
> equivalence, I find that an odd-limit condition is implied, which
> agrees perfectly with Partch's independent finding.

_____________________________________________________

J Gill: Here is what concerns me about "Tenney's" (harmonic
distance?, or) "height" ('TH'), as it relates to attempts to
characterize scale interval ratios in a manner commensurate with the
locations in frequency at which harmonics of simultaneously sounded
complex (fundamental plus harmonics) tones can be mathematically
predicted to "coincide" (or "overlap", or by virtue of their
proximities in pitch "interplay").

In the case where a "complex" (Tone 1) of reference is numerically
expressed as existing at a frequency of 1:1, and the frequency of a
second "complex" (Tone 2) is a rational number expressed in lowest
terms, and one wishes to determine the lowest common multiple ('LCM')
of the two frequenciy values (in order to determine the lowest
frequency at which harmonics of Tone 1 and Tone 2 will "coincide"):

(1) For 3:2 existing simultaneously with 1:1

3:2--[6:2]

1:1---2:1--[3:1]

(Unless I have interpreted something incorrectly in applying 'TH'),
it seems that the 'TH' of the *interval* existing between 1:1 and 3:2
(3/2) would be said to have a 'TH' value = 6.

Yet we can see by inspection (of the lowest frequency at which
[integer multiple] harmonics of the fundamental frequencies of Tone 1
and Tone 2 coincide, which is enclosed in [brackets] in the diagram
above), that this occurs at a frequency value = 3.

In fact, the numerical value of the denominators of the fundamental
frequencies 1:1 and 3:1 (and the numerical value of the denominator
of the 3/2 interval between these two fundamental frequencies)
appears to be INSIGNIFICANT in the determination of this lowest
frequency of "coincidence" of the harmonics of the fundamental
frequencies of Tone 1 and Tone 2.

In addition to harmonics (on the premise that they are specifically
present in both of the "complex" tones) "coinciding" at the frequency
which corresponds to the numerical 'LCM' of the two fundamental
frequency values, such "coincidences" will *also* occur at_all_
integer_multiples of this lowest frequency of "coincidence" of the
harmonics of the fundamental frequencies of Tone 1 and Tone 2.

(2) For 4:3 existing simultaneously with 1:1

4:3---8:3--[12/3]

1:1---2:1---3:1--[4:1]

it seems that the 'TH' of the *interval* existing between 1:1 and 4:3
(4/3) would be said to have a 'TH' value = 12.

Yet we can see by inspection (of the lowest frequency at which
[integer multiple] harmonics of the fundamental frequencies of Tone 1
and Tone 2 coincide, which is enclosed in [brackets] in the diagram
above), that this occurs at a frequency value = 4.

Similarly, for some other frequencies described by rational values
(listed for the "interval", in this case equal to the frequency of
Tone 2):

INTERVAL----'TH'----'LCM'

5/4----------20-------5
6/5----------30-------6
7/6----------42-------7
8/7----------56-------8
9/8----------72-------9
10/9---------90------10

16/15-------240------16

Therefore, the 'LCM' is equal to ('TH')/DENOMINATOR (or, restated,
the denominator is INSIGNIFICANT).

We can see by inspection (of the lowest frequency at which [integer
multiple] harmonics of the fundamental frequencies of Tone 1 and Tone
2 coincide, which are enclosed in [brackets] in the diagrams above),
that this occurs at a frequency value which is derived *exclusively*
from values in the_numerator_only.

In the different case where neither of the frequencies to be compared
are equal to a value of 1:1, 'TH' calculates the 'LCM' exactly *only*
in the (special) case where the fundamental frequency of Tone 2 is
equal to a value of 1/(Tone 1). Examples would be:

[Tone 1]---[Tone 2]----'TI'--'LCM'

---2:3-------3:2---------6------6
---3:4-------4:3--------12-----12
---4:5-------5:4--------20-----20
---5:6-------6:5--------30-----30

And so on (for all comparisons for which the denominator values of
the ratios under consideration do not share any prime factors)...

It appears (to me), as a result of these results, that "Tenney's
height" ('TH') relates accurately to the "lowest common multiple"
('LCM') ONLY in the case where one considers a given frequency (the
pitch of "complex" Tone 1) sounded simultaneously along with a second
frequency (the pitch of "complex" Tone 2) which is (as a rational
number), numerically equal to the algebraic *inversion* of the value
of the pitch of "complex" tone 1 (representing musical "inversion"
around the 1:1 frequency).

Such a case [of a scale pitch being simultaneously sounded together
with a second scale pitch (which, itself, represents a
musical "inversion" of the first scale pitch)] appears (to me) to be
a fairly *rare* musical occurance in practice (though not
impossible). So, it seems (to me) that 'TH' is *not* (except in the
above described "special" as well as "unlikely" case) a useful
measure of the the lowest frequency at which [integer multiple]
harmonics of the fundamental frequencies of Tone 1 and Tone 2
coincide (the 'LCM').

In that the reason(s) we speak of "harmonies" between certain scale
pitches (I think) must certainly derive (at least in part) as a
result of the "coincidences" in the frequency (pitch) locations of
the various integer multiple harmonics of the fundamental frequencies
which are sounded simultaneously in the presentation of musical
chords, it is my belief that the 'LCM' (as opposed to "Tenney's
height" ('TH') is the more meaningful and useful "metric" to apply.

Regards, J Gill :)

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> Huh? But isn't that what Harry Partch was trying to do with his
> tonality diamond?? Isn't that what he was getting at??

Yes, yes, yes, yes, and goshdarnit, yes.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> > I myself do not find that the quantity which is commonly referred
> to
> > as "harmonic distance" [that is: (numerator) * (denominator)] to
be
> > an accurate "metric" of "harmonic coincidence". As a result, when
> > modelling the frequency locations of "harmonic coincidence"
> existing
> > between two (or more) scale intervals sounded simultaneously, I
> > utilize a different numerical method for determining such results.
>
>PE: Could you elaborate it?

JG: See message /tuning/topicId_31425.html#31534

> > > PE: In any case, you must be refering to the construction of
*scales*
> > > with this

JG: My goal in this is more along the lines of analyzing
the "harmonic" basis (if such does exist) leading to certain JI scale
interval choices which have been made by others. However, in defining
a "universe" of JI scale intervals within a central JI "gamut" of
(possible) choices of scale intervals - which are dictated
by "harmonic" considerations, "scale-builders" might possibly find
a "limited" selection from which to choose, and may not like what
they see [as a result of "harmonic coincidences" occuring primarily
only when utilizing certain (low valued integer) JI scale intervals].
The "universe" of possibilities which I describe is *not* finite.
However, I have yet to evaluate whether "useful" JI intervals
continue to occur within its members beyond a certain subgroup (which
looks to be 12 basic scale "degrees" plus 2 alternate scale interval
ratios also included ([existing as (playable) substitutes for 2 of
the 12 scale intervals in order to fulfill certain desired
conditions], resulting in a total of 14 "playable notes" available
per octave on whatever musical instrument might implement such tuning.

> > JG: I am seeking to define a central "gamut" of possible notes
> > (through the prime 7-limit)

>PE: Odd 7-limit, perhaps?

JG: The largest prime number utilized in order to generate (possible)
scale interval ratios existing within the "universe" of the
central "gamut" is the prime number 7, indeed.

> > (As I see
> > it), in order for a unified harmonic/melodic JI structure to
exist,
> > certain sets of JI intervals and the (melodic) implications of
> their
> > inclusion within certain (N-tone, where N is an even valued
number)
> > JI scales
>
>PE: Why should N be even?

JG:

>I would be *very* interested in hearing about your (or
> > their) thoughts and ideas regarding such relationships (of which,
> in
> > my very limited knowledge of, and exposure to, the work of
others,
> I
> > am not, as yet, aware of as existing).
>
> Have you read Partch's book?

JG: No, I don't have a copy...

> > It *is* about why certain JI intervals *may* [when a (complex)
note
> > is simultaneously sounded in unison with, and relative to, any
> > additional given (complex) interval(s) and associated timber(s)
> > existing within a central "gamut" of (potential) JI scale
> intervals],
> > demonstrate various levels of tonal affinity for each other, as
> well
> > as demonstrate definable similarities between certain (complex)
> > intervals in terms of their characteristic harmonic interactions
> with
> > other (complex) intervals which exist within the "universe" of
> > intervals included within a central "gamut" of (potential) JI
scale
> > intervals.
>
> You're telling me you haven't read Partch's book?

JG: Correct. I've seen a little bit about him (mostly from postings
and papers generated by members of the ATL. That's it.

Question (JG): When (or if) Partch applied his "numerary nexus"
concepts in regards to "harmonic coincidences" (existing between
harmonic multiples of the fundamental frequencies with values
described by rational numbers formed of prime numbers taken to
integer powers), did he, at any point, differentiate (or in some way
distinguish) between: (1) the existence of a "numerary nexus"
occuring in the *numerator* of a scale interval ratio; and (2) the
existence of a "numerary nexus" occuring in the *denominator* of a
scale interval ratio? If so, what did he have to say about that?

Curiously, J Gill

--- In tuning@y..., "unidala" <JGill99@i...> wrote:

> _____________________________________________________
>
>
> J Gill: Here is what concerns me about "Tenney's" (harmonic
> distance?, or) "height" ('TH'), as it relates to attempts to
> characterize scale interval ratios in a manner commensurate with
the
> locations in frequency at which harmonics of simultaneously sounded
> complex (fundamental plus harmonics) tones can be mathematically
> predicted to "coincide" (or "overlap", or by virtue of their
> proximities in pitch "interplay").
>
> In the case where a "complex" (Tone 1) of reference is numerically
> expressed as existing at a frequency of 1:1, and the frequency of a
> second "complex" (Tone 2) is a rational number expressed in lowest
> terms, and one wishes to determine the lowest common multiple
('LCM')
> of the two frequenciy values (in order to determine the lowest
> frequency at which harmonics of Tone 1 and Tone 2 will "coincide"):
>
>
> (1) For 3:2 existing simultaneously with 1:1
>
> 3:2--[6:2]
>
> 1:1---2:1--[3:1]
>
> (Unless I have interpreted something incorrectly in applying 'TH'),
> it seems that the 'TH' of the *interval* existing between 1:1 and
3:2
> (3/2) would be said to have a 'TH' value = 6.

log(6).

> Yet we can see by inspection (of the lowest frequency at which
> [integer multiple] harmonics of the fundamental frequencies of Tone
1
> and Tone 2 coincide, which is enclosed in [brackets] in the diagram
> above), that this occurs at a frequency value = 3.

Only in units of the lower pitch. In units of the upper pitch, the
frequency value = 2. Why should the lower pitch be the basis of
measurement?

> In fact, the numerical value of the denominators of the fundamental
> frequencies 1:1 and 3:1 (and the numerical value of the denominator
> of the 3/2 interval between these two fundamental frequencies)
> appears to be INSIGNIFICANT in the determination of this lowest
> frequency of "coincidence" of the harmonics of the fundamental
> frequencies of Tone 1 and Tone 2.

Well, we can discuss that further, but if you still insist on using
the numerator only, you end up in the "integer limit" world which
we've discussed extensively, and is not very different from the
Tenney/Partch world.

> In addition to harmonics (on the premise that they are specifically
> present in both of the "complex" tones) "coinciding" at the
frequency
> which corresponds to the numerical 'LCM' of the two fundamental
> frequency values, such "coincidences" will *also* occur at_all_
> integer_multiples of this lowest frequency of "coincidence" of the
> harmonics of the fundamental frequencies of Tone 1 and Tone 2.

Correct.
>
> (2) For 4:3 existing simultaneously with 1:1
>
> 4:3---8:3--[12/3]
>
> 1:1---2:1---3:1--[4:1]
>
> it seems that the 'TH' of the *interval* existing between 1:1 and
4:3
> (4/3) would be said to have a 'TH' value = 12.

Log(12).
>
> Yet we can see by inspection (of the lowest frequency at which
> [integer multiple] harmonics of the fundamental frequencies of Tone
1
> and Tone 2 coincide, which is enclosed in [brackets] in the diagram
> above), that this occurs at a frequency value = 4.

Only in units of the lower tone. In units of the upper tone, it's
three. Why insist on using units of the lower tone and not of the
upper tone?

> In the different case where neither of the frequencies to be
compared
> are equal to a value of 1:1, 'TH' calculates the 'LCM' exactly
*only*
> in the (special) case where the fundamental frequency of Tone 2 is
> equal to a value of 1/(Tone 1). Examples would be:
>
> [Tone 1]---[Tone 2]----'TI'--'LCM'
>
> ---2:3-------3:2---------6------6

Whoa! Whoa! Whoa! :)

What is TI?

What is 6?

The ratio formed by a tone 2/3 and a tone 3/2 is 9:4. The Tenney
harmonic distance is log(36), and the "integer limit" is 9.

Where did the train fall off the tracks? :)

> It appears (to me), as a result of these results, that "Tenney's
> height" ('TH') relates accurately to the "lowest common multiple"
> ('LCM') ONLY in the case where one considers a given frequency (the
> pitch of "complex" Tone 1) sounded simultaneously along with a
second
> frequency (the pitch of "complex" Tone 2) which is (as a rational
> number), numerically equal to the algebraic *inversion* of the
value
> of the pitch of "complex" tone 1 (representing musical "inversion"
> around the 1:1 frequency).

Actually the 'TH' always agrees exactly with the LCM when the
frequency is measured in units of _the implied fundamental_ -- or
when the _wave period_ is measured in units of the _lowest common
overtone_.

--- In tuning@y..., "unidala" <JGill99@i...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning@y..., "unidala" <JGill99@i...> wrote:
> >
> > > I myself do not find that the quantity which is commonly
referred
> > to
> > > as "harmonic distance" [that is: (numerator) * (denominator)]
to
> be
> > > an accurate "metric" of "harmonic coincidence". As a result,
when
> > > modelling the frequency locations of "harmonic coincidence"
> > existing
> > > between two (or more) scale intervals sounded simultaneously, I
> > > utilize a different numerical method for determining such
results.
> >
> >PE: Could you elaborate it?
>
> JG: See message /tuning/topicId_31425.html#31534

The Prime Duodala puzzle? So you're presenting it as a puzzle?

> The "universe" of possibilities which I describe is *not* finite.
> However, I have yet to evaluate whether "useful" JI intervals
> continue to occur within its members beyond a certain subgroup

The same JI intervals appear over and over again, infinitely, as far
as you want to do in the lattice. So I can't see why you'd doubt this.

> > Have you read Partch's book?
>
> JG: No, I don't have a copy...

I'd encourage you to get one, and to spend some time reading it very
carefully . . . and very critically of course!
>
> Question (JG): When (or if) Partch applied his "numerary nexus"
> concepts in regards to "harmonic coincidences" (existing between
> harmonic multiples of the fundamental frequencies with values
> described by rational numbers formed of prime numbers taken to
> integer powers), did he, at any point, differentiate (or in some
way
> distinguish) between: (1) the existence of a "numerary nexus"
> occuring in the *numerator* of a scale interval ratio; and (2) the
> existence of a "numerary nexus" occuring in the *denominator* of a
> scale interval ratio? If so, what did he have to say about that?

He didn't focus on "harmonic coincidences" between upper partials,
but he did call chords with a numerary nexus in the
numerator "Undertone series" or "Utonal" chords, and those with a
numerary nexus in the denominatior "Overtone series" or "Otonal"
chords. We've discussed a lot on this list that, while "Utonal"
chords have "total harmonic coincidences" earlier and more often
than "Otonal" chords, "Otonal" chords are generally perceived as more
consonant than "Utonal" chords.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> > _____________________________________________________
> >
> >
> > J Gill: Here is what concerns me about "Tenney's" (harmonic
> > distance?, or) "height" ('TH'), as it relates to attempts to
> > characterize scale interval ratios in a manner commensurate with
> the
> > locations in frequency at which harmonics of simultaneously
sounded
> > complex (fundamental plus harmonics) tones can be mathematically
> > predicted to "coincide" (or "overlap", or by virtue of their
> > proximities in pitch "interplay").
> >
> > In the case where a "complex" (Tone 1) of reference is
numerically
> > expressed as existing at a frequency of 1:1, and the frequency of
a
> > second "complex" (Tone 2) is a rational number expressed in
lowest
> > terms, and one wishes to determine the lowest common multiple
> ('LCM')
> > of the two frequenciy values (in order to determine the lowest
> > frequency at which harmonics of Tone 1 and Tone 2
will "coincide"):
> >
> >
> > (1) For 3:2 existing simultaneously with 1:1
> >
> > 3:2--[6:2]
> >
> > 1:1---2:1--[3:1]
> >
> > (Unless I have interpreted something incorrectly in
applying 'TH'),
> > it seems that the 'TH' of the *interval* existing between 1:1 and
> 3:2
> > (3/2) would be said to have a 'TH' value = 6.
>
> log(6).

JG: Correct, Paul. I was just describing it in linear (arithmetic)
terms (being in the "integer harmonic multiple" business), as oppose
to geometrically (as in the more common unit octave description).

> > Yet we can see by inspection (of the lowest frequency at which
> > [integer multiple] harmonics of the fundamental frequencies of
Tone
> 1
> > and Tone 2 coincide, which is enclosed in [brackets] in the
diagram
> > above), that this occurs at a frequency value = 3.

>PE: Only in units of the lower pitch.
>In units of the upper pitch, the
> frequency value = 2. Why should the lower pitch be the basis of
> measurement?

JG: My goal is to have all numerical results (whether they involve
frequencies 1:1 and 3:2, or frequencies 1:1 and 2:3) of the 'LCM'
calculation remain referenced to a single reference fundamental
frequency and its harmonics occuring at integer multiples existing
*above* that reference frequency (of 1), in order that the results
tables will have a commonly based reference point of comparison.
In the first case directly above I consider the 'LCM' as 3 (relative
to a value of 1, and equal to the 3rd harmonic of 1), and in the
latter case, I consider the 'LCM' as 2 (relative to a value of 1, and
equal to the 2nd harmonic of 1). Your question is well taken, and
prompts me to (more accurately) refer to the operation as the "lowest
common multiple of the reference (frequency; pitch)" ('LCMR', or
something). Do you see conceptual error in proceeding this way?

> > In fact, the numerical value of the denominators of the
fundamental
> > frequencies 1:1 and [CORRECTION = 3:2] (and the numerical value
of the denominator
> > of the 3/2 interval between these two fundamental frequencies)
> > appears to be INSIGNIFICANT in the determination of this lowest
> > frequency of "coincidence" of the harmonics of the fundamental
> > frequencies of Tone 1 and Tone 2.
>
>PE: Well, we can discuss that further, but if you still insist on
>using
> the numerator only,

JG: The numerator is all that (to me) has appeared (so far) to be
necessary in pursuing my ideas. :)

>PE: you end up in the "integer limit" world which
> we've discussed extensively,

JG: I guess I might have some reading to do? I "plugged in" the
term "integer limit" into the Yahoo ATL search "engine", and (of
course) I'm getting all posts including *either "integer" or "limit",
but I did not see the phrase "integer limit" in any of the message
headers... Any specific references (or search terms)?

>PE: and is not very different from the
> Tenney/Partch world.

> > In addition to harmonics (on the premise that they are
specifically
> > present in both of the "complex" tones) "coinciding" at the
> frequency
> > which corresponds to the numerical 'LCM' of the two fundamental
> > frequency values, such "coincidences" will *also* occur at_all_
> > integer_multiples of this lowest frequency of "coincidence" of
the
> > harmonics of the fundamental frequencies of Tone 1 and Tone 2.

>PE: Correct.

> > (2) For 4:3 existing simultaneously with 1:1
> >
> > 4:3---8:3--[12/3]
> >
> > 1:1---2:1---3:1--[4:1]
> >
> > it seems that the 'TH' of the *interval* existing between 1:1 and
> 4:3
> > (4/3) would be said to have a 'TH' value = 12.

>PE: Log(12).

JG: Correct.

> > Yet we can see by inspection (of the lowest frequency at which
> > [integer multiple] harmonics of the fundamental frequencies of
Tone
> 1
> > and Tone 2 coincide, which is enclosed in [brackets] in the
diagram
> > above), that this occurs at a frequency value = 4.
>
> Only in units of the lower tone. In units of the upper tone, it's
> three. Why insist on using units of the lower tone and not of the
> upper tone?

JG: See above explanation responding to your previous "Why should the
lower pitch be the basis of measurement?

> > In the different case where neither of the frequencies to be
> compared
> > are equal to a value of 1:1, 'TH' calculates the 'LCM' exactly
> *only*
> > in the (special) case where the fundamental frequency of Tone 2
is
> > equal to a value of 1/(Tone 1). Examples would be:
> >
> > [Tone 1]---[Tone 2]----'TI'--'LCM'
> >
> > ---2:3-------3:2---------6------6
>
>PE: Whoa! Whoa! Whoa! :)
>
> What is TI?

JG: A typo. Should be 'TH'.

>PE: What is 6?

JG: 6 is the numerical value of the (newly described, thanks to your
question) "lowest common multiple of the reference (frequency;
pitch)" ('LCMR').

>PE: The ratio formed by a tone 2/3 and a tone 3/2 is 9:4. The Tenney
> harmonic distance is log(36)

JG: I see the "error of my ways" in the table above <|:o

>PE: and the "integer limit" is 9.

JG: "Integer limit", then, as the value of the numerator (only)?

> Where did the train fall off the tracks? :)

JG: Sorry, went "lucid" there for a minute... So, in the interest
of "no BS", I will "SNIP" following tainted text forthwith!!!

>PE: Actually the 'TH' always agrees exactly with the LCM when the
> frequency is measured in units of _the implied fundamental_

JG: Could you give me a more specific definition of what you mean
directly above by the phrase "implied fundamental"? :)

>PE: -- or
> when the _wave period_

JG: You do mean "period" (not it's algebraic inverse of "frequency")?

>PE: is measured in units of the _lowest common
> overtone_.

JG: So you are saying here that - once one has determined the
numerical (period, and not frequency?) value of the "lowest common
overtone", ... then (and only then) does one know at what precise
pitch the value of the "Tenney height" ('HT') is intended to be
*referenced* to??? I'm confused... Am I making this too complex?
Please, if you will, advise.

Sincerely, J Gill

--- In tuning@y..., "unidala" <JGill99@i...> wrote:

> JG: My goal is to have all numerical results (whether they involve
> frequencies 1:1 and 3:2, or frequencies 1:1 and 2:3) of the 'LCM'
> calculation remain referenced to a single reference fundamental
> frequency

But what if the perfect fifth interval occurs between some other two
frequencies? Then you'd have a different 'LCM' measure for it --
sometimes even an infinite one -- you wouldn't want that, would you?

Anyway, I understand you wishing to reference the calculation to a
single reference fundamental frequency, but in your two examples
above, you're referencing the lower frequency in the first example,
and the higher frequency in the second example. That's
hardly "putting everything on common footing". How about referencing
the _geometric middle_ betweeen the two frequencies instead?

> and its harmonics occuring at integer multiples existing
> *above* that reference frequency (of 1), in order that the results
> tables will have a commonly based reference point of comparison.

What about the pair of pitches 4/5 and 6/5, or 4/13 and 6/13, or 4/pi
and 6/pi, for example? They also form a 3:2 interval, and are just as
consonant with one another as, say, 2/3 with 1/1, or 1/1 with 3/2.

> In the first case directly above I consider the 'LCM' as 3
(relative
> to a value of 1, and equal to the 3rd harmonic of 1), and in the
> latter case, I consider the 'LCM' as 2 (relative to a value of 1,
and
> equal to the 2nd harmonic of 1).

So a _lower_ perfect fifth is more consonant than a _higher_ perfect
fifth? That doesn't seem to make much sense to me.

> Your question is well taken, and
> prompts me to (more accurately) refer to the operation as
the "lowest
> common multiple of the reference (frequency; pitch)" ('LCMR', or
> something). Do you see conceptual error in proceeding this
way?

Yes. However, a true LCM (least common multiple) approach will just
lead you back exactly to 'TH', as I tried to indicate before. The
ratio between 1/1 and 3/2, or between 2/3 and 1/1, or between 4/13
and 6/13, is a 3:2 ratio, and the LCM of 3 and 2 is 6.

> >PE: you end up in the "integer limit" world which
> > we've discussed extensively,
>
> JG: I guess I might have some reading to do? I "plugged in" the
> term "integer limit" into the Yahoo ATL search "engine", and (of
> course) I'm getting all posts including *either "integer"
or "limit",
> but I did not see the phrase "integer limit" in any of the message
> headers... Any specific references (or search terms)?

Well, not really, I think we can just talk through this stuff.

> >PE: and is not very different from the
> > Tenney/Partch world.

For example, a Farey series describes the contents of a "harmonic
sphere" around 1/1, in the integer limit world.
>
> >PE: The ratio formed by a tone 2/3 and a tone 3/2 is 9:4. The
Tenney
> > harmonic distance is log(36)
>
> JG: I see the "error of my ways" in the table above <|:o
>
> >PE: and the "integer limit" is 9.
>
> JG: "Integer limit", then, as the value of the numerator (only)?

Right, if all ratios are expressed as improper fractions . . .
although "integer limit" is an ambiguous term, and can also mean "the
highest numerator in a set of ratios".

> >PE: Actually the 'TH' always agrees exactly with the LCM when the
> > frequency is measured in units of _the implied fundamental_
>
> JG: Could you give me a more specific definition of what you mean
> directly above by the phrase "implied fundamental"? :)

The "virtual pitch", or the highest possible "missing root" such that
if there were a tone there, it would have harmonics right were the
tones in question are.

> >PE: -- or
> > when the _wave period_
>
> JG: You do mean "period" (not it's algebraic inverse
of "frequency")?

Yes.

> >PE: is measured in units of the _lowest common
> > overtone_.
>
> JG: So you are saying here that - once one has determined the
> numerical (period, and not frequency?) value of the "lowest common
> overtone", ... then (and only then) does one know at what precise
> pitch the value of the "Tenney height" ('HT') is intended to be
> *referenced* to???

No -- there are at least three ways to reference it.

(1) You can reference it to the frequency of the "implied
fundamental", which you therefore call "1" -- and then the LCM of the
frequencies of the two notes equals the 'TH'.

(2) You can reference it to the period of the lowest common overtone,
which you therefore call "1" -- and then the LCM of the periods of
the two notes equals the 'TH'.

(3) Originally, you were referencing the frequency of the lowest
common overtone, but then (seemingly inconsistently) the higher. If
you reference the geometric mean of the two notes by calling _that_
1, the frequency of the lowest common overtone will come out as equal
to the square root of 'TH'.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
> > --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> > > --- In tuning@y..., "unidala" <JGill99@i...> wrote:
> > >
> > > > I myself do not find that the quantity which is commonly
> referred
> > > to
> > > > as "harmonic distance" [that is: (numerator) * (denominator)]
> to
> > be
> > > > an accurate "metric" of "harmonic coincidence". As a result,
> when
> > > > modelling the frequency locations of "harmonic coincidence"
> > > existing
> > > > between two (or more) scale intervals sounded simultaneously,
I
> > > > utilize a different numerical method for determining such
> results.
> > >
> > >PE: Could you elaborate it?

> >
> > JG: See message /tuning/topicId_31425.html#31534

JG:CORRECTION(S.B. /tuning/topicId_31418.html#31532)

>PE: The Prime Duodala puzzle? So you're presenting it as a puzzle?

JG: Sorry to direct you to the wrong link, there. I've revised
the "Prime Duodala puzzle" post (to Rev 1), in order to clarify and
add relevant information. I, myself, do not like to spend earnest
hours on researching and communicating information, only to feel that
my time was spent on trying hard to understand unclear information...

Some of my thoughts surrounding the concepts I am working on, as well
as the full compliment of diagrams [which represent (hopefully) the
most concentrated and clear presentation of my ideas], are
still "works-in-progress". I apologize if my enthusiasm in making
some early "partial" (bad pun) presentations of some (but not all) of
what is in my mind should become an irritation for others.

My schedule is fairly busy with other obligations, and I have had a
brief (but transitory) time period in the last week's time where I
have been able to find some time to converse about this stuff in
posts. Unfortunately, it has not been enough time within which to
compile, edit, and diagram the concepts which I (in the future) hope
to be able to present in a more unified and coherent form. :)

No person should feel any obligation of any sort, other than the free
exercise of their own curiosity, to stare at "The Prime Duodala" and
wonder. I am happy to clarify what I am able to, but am waiting until
I have generated all the diagrams (and clearly explained those
diagrams in an accompanying explanatory text) to present in full.

Sincerely, J Gill

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

/tuning/topicId_31418.html#31538

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > Huh? But isn't that what Harry Partch was trying to do with his
> > tonality diamond?? Isn't that what he was getting at??
>
> Yes, yes, yes, yes, and goshdarnit, yes.

Well, gee... it's nice to know that I've learned at least *something*
sitting on these lists! :)

JP

Paul,

Sorry about the notational "180 deg phase shift" on my (and not your)
part in these recent threads - where I (in a attempt to conform to
common practice!) have "flipped" the ASCII characters (used ":"
for "pitches", and "/" for "intervals"). I will modify my notation in
this full post to your common form below, and remain phase-locked.. :)

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:

> > and its harmonics occuring at integer multiples existing
> > *above* that reference frequency (of 1/1), in order that the
results
> > tables will have a commonly based reference point of comparison.

>PE: What about the pair of pitches 4/5 and 6/5,

JG: 'LCMR' ("lowest common multiple referenced to 1/1") = 12/5.

>PE: or 4/13 and 6/13,

JG: 'LCMR' = 12/13

>PE: or 4/pi
> and 6/pi, for example?

JG: 'LCMR' = 12/1 (because the denominators of the two ratios do
*not* share a common prime factor)

>PE: They also form a 3:2 interval, and are just as
> consonant with one another as, say, 2/3 with 1/1, or 1/1 with 3/2.

JG: Well, the 'LCMR's (respectively), are equal 2/1 and 3/1, because -
that is the frequency (*relative* to the pitch of 1/1) at which
the "lowest frequency of harmonic coincidence" appears in a (linear
horizontal scale) frequency spectrum display... Seems well grounded.

> > In the first case directly above I consider the 'LCM' as 3/1
> (relative
> > to a value of 1/1, and equal to the 3rd harmonic of 1/1), and in
>the
> > latter case, I consider the 'LCM' as 2/1 (relative to a value of
>1/1,
> and
> > equal to the 2nd harmonic of 1/1).
>
>PE: So a _lower_ perfect fifth is more consonant than a _higher_
>perfect
> fifth? That doesn't seem to make much sense to me.

JG: All the 'LCMR' says is that (harmonically, and hence spectrally)
it's value is a smaller number in the case of 2/3 compared to 1/1
than it is in the case of 3/2 compared to 1/1. This indicates that
a "harmonic series of coinciding inter-tone harmonics" does (in the
case of 2/3 compared to 1/1) begin at a *lower* frequency (2/1) than
is the case where where 3/2 is compared to 1/1 (at frequency = 3/1).

JG: I'm not at all certain (and have not, I hope, left you with the
impression) that I have defined what constitutes the (far more
complicated, hotly debated permutations of a) "consonance" (concept)
is fully_explained, or otherwise somehow directly_related_to, this
determination of an 'LCM' of two rational numbers which referenced to
a rational number equal to 1/1 ( the 'LCMR').

With that in mind, one of the things that has intrigued me about the
prospect of the (octave specific) world of calculating composite
spectrums of "complex" tones is (at least) the possibility that - a
musical JI 4th (2/3) played in unison *below* the 1/1 MAY WELL (in a
spectral, hence timbral characteristic) SOUND DIFFERENT than a
musical JI 5th (3/2) played in unison above the 1/1. My primary aim
in developing a system of examining "harmonic" relationships is to
(wherever possible) avoid making personal judgements about the
desirability of combining various scale pitches in chords, focusing
more upon what the mathematical structures yield as a JI "harmonic
template", a "universe" from which some choices from the
central "gamut" may prove to be more usable than many of the others.

I am more interested in exploring some of the "whys" and "why nots"
of the various choices of JI intervals (without regard to
temperament, or other methods by which the JI "musical structure" may
be altered in the interest of "moving around on the keyboard"
successfully). Whether such insights would provide lots of scale
pitches with which to construct N-tone scales where N approaches
infinty (!), I do not know. I see the 'LCMR' as more of an "analysis"
tool than as a "synthesis tool"...

As to "consonance", "concordance", and "tonal affinities", simply
knowing where all the harmonics "coincide" may contribute to (but not
universally define or encompass) the characteristics of
such "complex" combinations of JI scale pitches, where the "spectral
forest" in full of many (time-variant) "trees", only some of
which "coincide". We would probably all like to know the "full" story!

Terhardt (on his website) is fairly brief regarding his fascinating
(but somewhat vague) "tonal affinity" concepts. Do know of any other
treatments of his "tonal affinity" concepts available by other means?

> > Your question is well taken, and
> > prompts me to (more accurately) refer to the operation as
> the "lowest
> > common multiple of the reference (frequency; pitch)" ('LCMR', or
> > something). Do you see conceptual error in proceeding this
> way?
>
>PE: Yes. However, a true LCM (least common multiple) approach will
>just
> lead you back exactly to 'TH', as I tried to indicate before. The
> ratio between 1/1 and 3/2, or between 2/3 and 1/1, or between 4/13
> and 6/13, is a 3:2 ratio, and the LCM of 3 and 2 is 6.
>
> > >PE: you end up in the "integer limit" world which
> > > we've discussed extensively,
> >
> > JG: I guess I might have some reading to do? I "plugged in" the
> > term "integer limit" into the Yahoo ATL search "engine", and (of
> > course) I'm getting all posts including *either "integer"
> or "limit",
> > but I did not see the phrase "integer limit" in any of the
>message
> > headers... Any specific references (or search terms)?
>
> Well, not really, I think we can just talk through this stuff.
>
> > >PE: and is not very different from the
> > > Tenney/Partch world.
>
> For example, a Farey series describes the contents of a "harmonic
> sphere" around 1/1, in the integer limit world.
> >
> > >PE: The ratio formed by a tone 2/3 and a tone 3/2 is 9:4. The
> Tenney
> > > harmonic distance is log(36)
> >
> > JG: I see the "error of my ways" in the table above <|:o
> >
> > >PE: and the "integer limit" is 9.
> >
> > JG: "Integer limit", then, as the value of the numerator (only)?
>
> Right, if all ratios are expressed as improper fractions . . .
> although "integer limit" is an ambiguous term, and can also
>mean "the
> highest numerator in a set of ratios".
>
> > >PE: Actually the 'TH' always agrees exactly with the LCM when
>the
> > > frequency is measured in units of _the implied fundamental_
> >
> > JG: Could you give me a more specific definition of what you mean
> > directly above by the phrase "implied fundamental"? :)
>
> The "virtual pitch", or the highest possible "missing root" such
>that
> if there were a tone there, it would have harmonics right were the
> tones in question are.
>
> > >PE: -- or
> > > when the _wave period_
> >
> > JG: You do mean "period" (not it's algebraic inverse
> of "frequency")?
>
> Yes.
>
> > >PE: is measured in units of the _lowest common
> > > overtone_.
> >
> > JG: So you are saying here that - once one has determined the
> > numerical (period, and not frequency?) value of the "lowest
common
> > overtone", ... then (and only then) does one know at what precise
> > pitch the value of the "Tenney height" ('HT') is intended to be
> > *referenced* to???
>
> No -- there are at least three ways to reference it.
>
> (1) You can reference it to the frequency of the "implied
> fundamental", which you therefore call "1" -- and then the LCM of
>the
> frequencies of the two notes equals the 'TH'.
>
> (2) You can reference it to the period of the lowest common
>overtone,
> which you therefore call "1" -- and then the LCM of the periods of
> the two notes equals the 'TH'.
>
> (3) Originally, you were referencing the frequency of the lowest
> common overtone, but then (seemingly inconsistently) the higher. If
> you reference the geometric mean of the two notes by calling _that_
> 1, the frequency of the lowest common overtone will come out as
>equal
> to the square root of 'TH'.

JG: You have here given me a good bit to "chew on" here (and I
appreciate your taking your time to consider some of these ideas, and
offer information and feedback which is valuable to me. Thank you.
Will study this post, and probably have a question or two for you!

Best Regards, J Gill

> > --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> > > and its harmonics occuring at integer multiples existing
> > > *above* that reference frequency (of 1/1), in order that the
> results
> > > tables will have a commonly based reference point of comparison.
>
> >PE: What about the pair of pitches 4/5 and 6/5,
>
> JG: 'LCMR' ("lowest common multiple referenced to 1/1") = 12/5.
>
> >PE: or 4/13 and 6/13,
>
> JG: 'LCMR' = 12/13
>
> >PE: or 4/pi
> > and 6/pi, for example?
>
> JG: 'LCMR' = 12/1 (because the denominators of the two ratios do
> *not* share a common prime factor)
>
> >PE: They also form a 3:2 interval, and are just as
> > consonant with one another as, say, 2/3 with 1/1, or 1/1 with 3/2.
>
> JG: Well, the 'LCMR's (respectively), are equal 2/1 and 3/1,
because -
> that is the frequency (*relative* to the pitch of 1/1) at which
> the "lowest frequency of harmonic coincidence" appears in a (linear
> horizontal scale) frequency spectrum display... Seems well grounded.

What??? This seems to make no sense at all! Let me give you more
examples:

4/e and 6/e
24/25 and 36/25
Shall I go on?

> JG: All the 'LCMR' says is that (harmonically, and hence
spectrally)
> it's value is a smaller number in the case of 2/3 compared to 1/1
> than it is in the case of 3/2 compared to 1/1. This indicates that
> a "harmonic series of coinciding inter-tone harmonics" does (in the
> case of 2/3 compared to 1/1) begin at a *lower* frequency (2/1)
than
> is the case where where 3/2 is compared to 1/1 (at frequency = 3/1).

True . . . but your 4/pi : 6/pi example would _not_ seem to conform
to this logic!
>
> With that in mind, one of the things that has intrigued me about
the
> prospect of the (octave specific) world of calculating composite
> spectrums of "complex" tones is (at least) the possibility that - a
> musical JI 4th (2/3) played in unison *below* the 1/1 MAY WELL (in
a
> spectral, hence timbral characteristic) SOUND DIFFERENT than a
> musical JI 5th (3/2) played in unison above the 1/1.

Well, if you have a fixed 1/1, it's lower and, if too low, _less_
consonant . . . But what if you _haven't_ designated a 1/1? Shouldn't
you be able to evaluate intervals _as intervals_, without reference
to any putative choice of tonal center?

Let's say you say no and stick with your idea of relating everything
back to a fixed 1/1 (you'll still have to modify your approach with
pi, etc.) You'll end up finding that all members of an infinite
undertone series under 1/1 (i.e., 1/1, 1/2, 1/3, 1/4, 1/5, etc.) have
an 'LCMR' of 1 when taken two, or more, at a time. This would seem to
be the "ultimate consonant scale" under your rule. But if you listen
to the sound two low notes, say 1/10 and 1/11, make when played
together, you might reconsider . . .

--- In tuning@y..., "unidala" <JGill99@i...> wrote:

/tuning/topicId_31418.html#31563

>
> With that in mind, one of the things that has intrigued me about
the
> prospect of the (octave specific) world of calculating composite
> spectrums of "complex" tones is (at least) the possibility that - a
> musical JI 4th (2/3) played in unison *below* the 1/1 MAY WELL (in
a
> spectral, hence timbral characteristic) SOUND DIFFERENT than a
> musical JI 5th (3/2) played in unison above the 1/1. My primary aim
> in developing a system of examining "harmonic" relationships is to
> (wherever possible) avoid making personal judgements about the
> desirability of combining various scale pitches in chords, focusing
> more upon what the mathematical structures yield as a JI "harmonic
> template", a "universe" from which some choices from the
> central "gamut" may prove to be more usable than many of the others.
>

I believe, J., that the difference between the otonal and utonal
intervals and chords was discussed here previously by Paul Erlich and
others. The utonal chords have a peculiar "wobble" that is hard to
explain, but *definitely* different from the otonal ones. And it
isn't exactly a sense of "beating" either. It's something rather
peculiar and almost "mysterious." There are some examples on
the "Tuning Lab" website that you might be interested in listening
to. The examples at the very top illustrate some of these
distinctions...

http://artists.mp3s.com/artists/140/tuning_lab.html

best,

JP

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "unidala" <JGill99@i...> wrote:
>
> /tuning/topicId_31418.html#31563
>
> >
> > With that in mind, one of the things that has intrigued me about
> the
> > prospect of the (octave specific) world of calculating composite
> > spectrums of "complex" tones is (at least) the possibility that - a
> > musical JI 4th (2/3) played in unison *below* the 1/1 MAY WELL (in
> a
> > spectral, hence timbral characteristic) SOUND DIFFERENT than a
> > musical JI 5th (3/2) played in unison above the 1/1. My primary aim
> > in developing a system of examining "harmonic" relationships is to
> > (wherever possible) avoid making personal judgements about the
> > desirability of combining various scale pitches in chords, focusing
> > more upon what the mathematical structures yield as a JI "harmonic
> > template", a "universe" from which some choices from the
> > central "gamut" may prove to be more usable than many of the others.
> >
>
> I believe, J., that the difference between the otonal and utonal
> intervals and chords was discussed here previously by Paul Erlich and
> others.

Sorry, Joseph, but I don't see how that's relevant here. Jeremy is
talking about two instances of a perfect fifth dyad. There's no
otonal/utonal distinction between the two -- they're identical up to a
transposition.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

/tuning/topicId_31418.html#31587

> >
> > I believe, J., that the difference between the otonal and utonal
> > intervals and chords was discussed here previously by Paul Erlich
and
> > others.
>
> Sorry, Joseph, but I don't see how that's relevant here. Jeremy is
> talking about two instances of a perfect fifth dyad. There's no
> otonal/utonal distinction between the two -- they're identical up
to a transposition.

Hi Paul...

Oh, I guess you're right in this case.... I was just wondering if he
had ever heard utonal and otonal approximations of the same chord and
the "warbling" that we noticed in the utonal (as distinct from
beating) as we had seen, I think, on Tuning Lab... (as a general
case...)

best,

Joseph